Browsing by Subject "Inverse Problem"
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Item A Hybrid Ensemble Kalman Filter for Nonlinear Dynamics(2011-02-22) Watanabe, ShingoIn this thesis, we propose two novel approaches for hybrid Ensemble Kalman Filter (EnKF) to overcome limitations of the traditional EnKF. The first approach is to swap the ensemble mean for the ensemble mode estimation to improve the covariance calculation in EnKF. The second approach is a coarse scale permeability constraint while updating in EnKF. Both hybrid EnKF approaches are coupled with the streamline based Generalized Travel Time Inversion (GTTI) algorithm for periodic updating of the mean of the ensemble and to sequentially update the ensemble in a hybrid fashion. Through the development of the hybrid EnKF algorithm, the characteristics of the EnKF are also investigated. We found that the limits of the updated values constrain the assimilation results significantly and it is important to assess the measurement error variance to have a proper balance between preserving the prior information and the observation data misfit. Overshooting problems can be mitigated with the streamline based covariance localizations and normal score transformation of the parameters to support the Gaussian error statistics. The swapping mean and mode estimation approach can give us a better matching of the data as long as the mode solution of the inversion process is satisfactory in terms of matching the observation trajectory. The coarse scale permeability constrained hybrid approach gives us better parameter estimation in terms of capturing the main trend of the permeability field and each ensemble member is driven to the posterior mode solution from the inversion process. However the WWCT responses and pressure responses need to be captured through the inversion process to generate physically plausible coarse scale permeability data to constrain hybrid EnKF updating. Uncertainty quantification methods for EnKF were developed to verify the performance of the proposed hybrid EnKF compared to the traditional EnKF. The results show better assimilation quality through a sequence of updating and a stable solution is demonstrated. The potential of the proposed hybrid approaches are promising through the synthetic examples and a field scale application.Item Comparative Deterministic and Probabilistic Modeling in Geotechnics: Applications to Stabilization of Organic Soils, Determination of Unknown Foundations for Bridge Scour, and One-Dimensional Diffusion Processes(2013-08-08) Yousefpour, NeginThis study presents different aspects on the use of deterministic methods including Artificial Neural Networks (ANNs), and linear and nonlinear regression, as well as probabilistic methods including Bayesian inference and Monte Carlo methods to develop reliable solutions for challenging problems in geotechnics. This study addresses the theoretical and computational advantages and limitations of these methods in application to: 1) prediction of the stiffness and strength of stabilized organic soils, 2) determination of unknown foundations for bridges vulnerable to scour, and 3) uncertainty quantification for one-dimensional diffusion processes. ANNs were successfully implemented in this study to develop nonlinear models for the mechanical properties of stabilized organic soils. ANN models were able to learn from the training examples and then generalize the trend to make predictions for the stiffness and strength of stabilized organic soils. A stepwise parameter selection and a sensitivity analysis method were implemented to identify the most relevant factors for the prediction of the stiffness and strength. Also, the variations of the stiffness and strength with respect to each factor were investigated. A deterministic and a probabilistic approach were proposed to evaluate the characteristics of unknown foundations of bridges subjected to scour. The proposed methods were successfully implemented and validated by collecting data for bridges in the Bryan District. ANN models were developed and trained using the database of bridges to predict the foundation type and embedment depth. The probabilistic Bayesian approach generated probability distributions for the foundation and soil characteristics and was able to capture the uncertainty in the predictions. The parametric and numerical uncertainties in the one-dimensional diffusion process were evaluated under varying observation conditions. The inverse problem was solved using Bayesian inference formulated by both the analytical and numerical solutions of the ordinary differential equation of diffusion. The numerical uncertainty was evaluated by comparing the mean and standard deviation of the posterior realizations of the process corresponding to the analytical and numerical solutions of the forward problem. It was shown that higher correlation in the structure of the observations increased both parametric and numerical uncertainties, whereas increasing the number of data dramatically decreased the uncertainties in the diffusion process.Item Nuclear magnetic resonance imaging and analysis for determination of porous media properties(Texas A&M University, 2007-04-25) Uh, JinsooAdvanced nuclear magnetic resonance (NMR) imaging methodologies have been developed to determine porous media properties associated with fluid flow processes. This dissertation presents the development of NMR experimental and analysis methodologies, called NMR probes, particularly for determination of porosity, permeability, and pore-size distributions of porous media while the developed methodologies can be used for other properties. The NMR relaxation distribution can provide various information about porous systems having NMR active nuclei. The determination of the distribution from NMR relaxation data is an ill-posed inverse problem that requires special care, but conventionally the problem has been solved by ad-hoc methods. We have developed a new method based on sound statistical theory that suitably implements smoothness and equality/inequality constraints. This method is used for determination of porosity distributions. A Carr-Purcell-Meiboom-Gill (CPMG) NMR experiment is designed to measure spatially resolved NMR relaxation data. The determined relaxation distribution provides the estimate of intrinsic magnetization which, in turn, is scaled to porosity. A pulsed-field-gradient stimulated-echo (PFGSTE) NMR velocity imaging experiment is designed to measure the superficial average velocity at each volume element. This experiment measures velocity number distributions as opposed to the average phase shift, which is conventionally measured, to suitably quantify the velocities within heterogeneous porous media. The permeability distributions are determined by solving the inverse problem formulated in terms of flow models and the velocity data. We present new experimental designs associated with flow conditions to enhance the accuracy of the estimates. Efforts have been put forth to further improve the accuracy by introducing and evaluating global optimization methods. The NMR relaxation distribution can be scaled to a pore-size distribution once the surface relaxivity is known. We have developed a new method, which avoids limitations on the range of time for which data may be used, to determine surface relaxivity by the PFGSTE NMR diffusion experiment.Item Scattered neutron tomography based on a neutron transport problem(Texas A&M University, 2005-11-01) Scipolo, VittorioTomography refers to the cross-sectional imaging of an object from either transmission or reflection data collected by illuminating the object from many different directions. Classical tomography fails to reconstruct the optical properties of thick scattering objects because it does not adequately account for the scattering component of the neutron beam intensity exiting the sample. We proposed a new method of computed tomography which employs an inverse problem analysis of both the transmitted and scattered images generated from a beam passing through an optically thick object. This inverse problem makes use of a computationally efficient, two-dimensional forward problem based on neutron transport theory that effectively calculates the detector readings around the edges of an object. The forward problem solution uses a Step-Characteristic (SC) code with known uncollided source per cell, zero boundary flux condition and Sn discretization for the angular dependence. The calculation of the uncollided sources is performed by using an accurate discretization scheme given properties and position of the incoming beam and beam collimator. The detector predictions are obtained considering both the collided and uncollided components of the incoming radiation. The inverse problem is referred as an optimization problem. The function to be minimized, called an objective function, is calculated as the normalized-squared error between predicted and measured data. The predicted data are calculated by assuming a uniform distribution for the optical properties of the object. The objective function depends directly on the optical properties of the object; therefore, by minimizing it, the correct property distribution can be found. The minimization of this multidimensional function is performed with the Polack Ribiere conjugate-gradient technique that makes use of the gradient of the function with respect to the cross sections of the internal cells of the domain. The forward and inverse models have been successfully tested against numerical results obtained with MCNP (Monte Carlo Neutral Particles) showing excellent agreements. The reconstructions of several objects were successful. In the case of a single intrusion, TNTs (Tomography Neutron Transport using Scattering) was always able to detect the intrusion. In the case of the double body object, TNTs was able to reconstruct partially the optical distribution. The most important defect, in terms of gradient, was correctly located and reconstructed. Difficulties were discovered in the location and reconstruction of the second defect. Nevertheless, the results are exceptional considering they were obtained by lightening the object from only one side. The use of multiple beams around the object will significantly improve the capability of TNTs since it increases the number of constraints for the minimization problem.Item Ultrasound-modulated optical tomography(Texas A&M University, 2004-09-30) Nam, HaewonUltrasound-modulated optical tomography is modeled by a linear integral equation and an inverse problem involving a diffusion equation in n spatial dimensions, n=2, 3. Based on measured data, the optical absorption coefficient ? is reconstructed inside of a given domain. We make a two-step mathematical model. First, we solve a linear integral equation. Assuming the energy fluence rate has been recovered from the previous equation, the absorption coefficient ? is then reconstructed by solving an inverse problem. Numerical experiments are presented for the case n=2. Two methods are used for the numerical experiments, gradient descent and levelset. At the end, advantages and disadvantages of those two methods are mentioned.